A090285 - OEIS

%I #33 Jun 12 2019 11:17:13

%S 1,1,1,2,4,1,5,15,7,1,14,56,37,10,1,42,210,176,68,13,1,132,792,794,

%T 392,108,16,1,429,3003,3473,2063,731,157,19,1,1430,11440,14893,10254,

%U 4395,1220,215,22,1,4862,43758,63004,49024,24465,8249,1886,282,25,1

%N Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(n,k)=0 if k>n, T(n,0) = A000108(n); T(n+1,k)= Sum_{j=0..n} T(n-j,k-1)*binomial(2j+1,j+1).

%C The matrix inverse starts

%C     1;

%C    -1,     1;

%C     2,    -4,     1;

%C    -4,    13,    -7,     1;

%C     8,   -38,    33,   -10,    1;

%C   -16,   104,  -129,    62,  -13,     1;

%C    32,  -272,   450,  -304,  100,   -16,   1;

%C   -64,   688, -1452,  1289, -590,   147, -19,   1;

%C   128, -1696,  4424, -4942, 2945, -1014, 203, -22, 1;

%C - _R. J. Mathar_, Mar 15 2013

%C Riordan array (c(x), x*c(x)^2/(1-x*c(x)^2)) where c(x) is the g.f. for the Catalan numbers (A000108). - _Philippe Deléham_, Jun 02 2013

%C The matrix inverse is the Riordan array ((1+x)/(1+2*x), x*(1+x)/(1+2*x)^2). - _Philippe Deléham_, Jan 26 2014

%H Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa <a href="https://doi.org/10.1155/2014/316535">Noncontiguous pattern containment in binary trees</a>, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Table 1.

%H Efrat Engel Shaposhnik, <a href="http://hdl.handle.net/1721.1/104603">Antichains of Interval Orders and Semiorders, and Dilworth Lattices of maximum size Antichains</a>, Massachusetts Institute of Technology, June 2016.

%F T(n, 1) = n*A000108(n) = A001791(n) .

%F T(n, 2) = 2^(2n-1) - binomial(2n+1, n) + binomial(2n-1, n-1) = A006419(n).

%p A090285 := proc(n,k)

%p     if k < 0 or k > n then

%p         0 ;

%p     elif k = 0 then

%p         A000108(n)

%p     else

%p         add(procname(n-1-j,k-1)*binomial(2*j+1,j+1),j=0..n-1) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Mar 15 2013

%t T[n_, k_] := T[n, k] = If[k == 0, CatalanNumber@ n, Sum[T[(n - 1) - j, k - 1] Binomial[2 j + 1, j + 1], {j, 0, n - 1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Jun 26 2017 *)

%Y Diagonals: A000108, A001791, A006419; A000012, A016777.

%Y See also A001700 for binomial(2n+1,n+1).

%K easy,nonn,tabl

%O 0,4

%A _Philippe Deléham_, Jan 24 2004